Preface -- Notation -- 0 Introduction -- 1 Preliminaries -- 1.1 Simple transformations and examples -- 1.1.1 Dirac-type systems as a subclass of canonical systems -- 1.1.2 Schrödinger systems as a subclass of canonical systems -- 1.1.3 Gauge transformations of the Dirac systems -- 1.2 S-nodes and Weyl functions -- 1.2.1 Elementary properties of S-nodes -- 1.2.2 Continual factorization -- 1.2.3 Canonical systems and representation of the S-nodes -- 1.2.4 Asymptotics of the Weyl functions, a special case -- 1.2.5 Factorization of the operators S -- 1.2.6 Weyl functions of Dirac and Schrödinger systems -- 2 Self-adjoint Dirac system: rectangular matrix potentials -- 2.1 Square matrix potentials: spectral and Weyl theories -- 2.1.1 Spectral and Weyl functions: direct problem -- 2.1.2 Spectral and Weyl functions: inverse problem -- 2.2 Weyl theory for Dirac system with a rectangularmatrix potential -- 2.2.1 Direct problem -- 2.2.2 Direct and inverse problems: explicit solutions -- 2.3 Recovery of the Dirac system: general case -- 2.3.1 Representation of the fundamental solution -- 2.3.2 Weyl function: high energy asymptotics -- 2.3.3 Inverse problem and Borg-Marchenko-type uniqueness theorem -- 2.3.4 Weyl function and positivity of S -- 3 Skew-self-adjoint Dirac system: rectangular matrix potentials -- 3.1 Direct problem -- 3.2 The inverse problem on a finite interval and semiaxis -- 3.3 System with a locally bounded potential -- 4 Linear system auxiliary to the nonlinear optics equation -- 4.1 Direct and inverse problems -- 4.1.1 Bounded potentials -- 4.1.2 Locally bounded potentials -- 4.1.3 Weyl functions -- 4.1.4 Some generalizations -- 4.2 Conditions on the potential and asymptotics of generalized Weyl (GW) functions -- 4.2.1 Preliminaries. Beals-Coifman asymptotics.

4.2.2 Inverse problem and Borg-Marchenko-type result -- 4.3 Direct and inverse problems: explicit solutions -- 5 Discretesystems -- 5.1 Discrete self-adjoint Dirac system -- 5.1.1 Dirac system and Szegö recurrence -- 5.1.2 Weyl theory: direct problems -- 5.1.3 Weyl theory: inverse problems -- 5.2 Discrete skew-self-adjoint Dirac system -- 5.3 GBDT for the discrete skew-self-adjoint Dirac system -- 5.3.1 Main results -- 5.3.2 The fundamental solution -- 5.3.3 Weyl functions: direct and inverse problems -- 5.3.4 Isotropic Heisenberg magnet -- 6 Integrable nonlinear equations -- 6.1 Compatibility condition and factorization formula -- 6.1.1 Main results -- 6.1.2 Proof of Theorem 6.1 -- 6.1.3 Application to the matrix "focusing" modified Korteweg-de Vries (mKdV) -- 6.1.4 Second harmonic generation: Goursat problem -- 6.2 Sine-Gordon theory in a semistrip -- 6.2.1 Complex sine-Gordon equation: evolution of the Weyl function and uniqueness of the solution -- 6.2.2 Sine-Gordon equation in a semistrip -- 6.2.3 Unbounded solutions in the quarter-plane -- 7 General GBDT theorems and explicit solutions of nonlinear equations -- 7.1 Explicit solutions of the nonlinear optics equation -- 7.2 GBDT for linear system depending rationally on z -- 7.3 Explicit solutions of nonlinear equations -- 8 Some further results on inverse problems and generalized Bäcklund-Darboux transformation (GBDT) -- 8.1 Inverse problems and the evolution of the Weyl functions -- 8.2 GBDT for one and several variables -- 9 Sliding inverse problems for radial Dirac and Schrödinger equations -- 9.1 Inverse and half-inverse sliding problems -- 9.1.1 Main definitions and results -- 9.1.2 Radial Schrödinger equation and quantum defect -- 9.1.3 Dirac equation and quantum defect -- 9.1.4 Proofs of Theorems 9.10 and 9.14.

9.1.5 Dirac system on a finite interval -- 9.2 Schrödinger and Dirac equations with Coulomb-type potentials -- 9.2.1 Asymptotics of the solutions: Schrödinger equation -- 9.2.2 Asymptotics of the solutions: Dirac system -- Appendices -- A General-type canonical system: pseudospectral and Weyl functions -- A.1 Spectral and pseudospectral functions -- A.1.1 Basic notions and results -- A.1.2 Description of the pseudospectral functions -- A.1.3 Potapov's inequalities and pseudospectral functions -- A.1.4 Description of the spectral functions -- A.2 Special cases -- A.2.1 Positivity-type condition -- A.2.2 Continuous analogs of orthogonal polynomials -- B Mathematical system theory -- C Krein's system -- D Operator identities corresponding to inverse problems -- D.1 Operator identity: the case of self-adjoint Dirac system -- D.2 Operator identity for skew-self-adjoint Dirac system -- D.3 Families of positive operators -- D.4 Semiseparable operators S -- D.5 Operators with D-difference kernels -- E Somebasictheorems -- Bibliography -- Index.